The prime number race and zeros of Dirichlet \(L\)-functions off the critical line. III (Q2871298)

As usual, \(\pi(x;q,a)\) denotes the number of primes \(p\leq x\) with \(p =a\) (mod \(q\)), where \((a,q)=1\). The prime number race refers to the relative magnitudes of the function \(\pi(x;q,a)\) for fixed \(q\) and varying \(a\). This paper assumes \(q\geq 3\) and treats reduced residues \(a, b\) modulo \(q\). It is shown \(\pi(x;q,a)<\pi(x;q,b)\) for a set of real \(x\) of asymptotic density 1 if there exist certain sets of zeros of Dirichlet \(L\)-functions off the critical line.